Initially, the numbers $1,3,4$ are written on a board. We do the following process repeatedly. Consider all of the numbers that can be obtained as the sum of $3$ distinct numbers written on the board and that aren't already written, and we write those numbers on the board. We repeat this process, until at a certain step, all of the numbers in that step are greater than $2024$. Determine all of the integers $1\leq k\leq 2024$ that were not written on the board.